The 'Arithmetic and Geometry' trimester, held at the Hausdorff Research Institute for Mathematics in Bonn, focussed on recent work on Serre's conjecture and on rational points on algebraic varieties. The resulting proceedings volume provides a modern overview of the subject for graduate students in arithmetic geometry and Diophantine geometry. It is also essential reading for any researcher wishing to keep abreast of the latest developments in the field. Highlights include Tim Browning's survey on applications of the circle method to rational points on algebraic varieties and Per Salberger's chapter on rational points on cubic hypersurfaces.
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The world's leading authorities describe the state of the art in Serre's conjecture and rational points on algebraic varieties.
| Preface |
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vii | |
| Introduction |
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ix | |
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| Trimester Seminar |
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xiii | |
| Workshop on Serre's conjecture |
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xxiii | |
| The research conference |
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xxviii | |
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1 Galois groups of local fields, Lie algebras and ramification |
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1 | (23) |
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2 A characterisation of ordinary modular eigenforms with CM |
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24 | (12) |
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3 Selmer complexes and p-adic Hodge theory |
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36 | (53) |
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4 A survey of applications of the circle method to rational points |
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89 | (25) |
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5 Arithmetic differential equations of Painleve VI type |
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114 | (25) |
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6 Differential calculus with integers |
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139 | (49) |
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7 Un calcul de groupe de Brauer et une application arithmetique |
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188 | (5) |
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Jean-Louis Colliot-Thelene |
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8 Connectedness of Hecke algebras and the Rayuela conjecture: a path to functoriality and modularity |
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193 | (24) |
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9 Big image of Galois representations and congruence ideals |
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217 | (38) |
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10 The skew-symmetric pairing on the Lubin--Tate formal module |
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255 | (9) |
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11 Equations in matrix groups and algebras over number fields and rings: prolegomena to a lowbrow noncommutative Diophantine geometry |
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264 | (19) |
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12 On the e-adic regulator as an ingredient of Iwasawa theory |
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283 | (35) |
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13 On a counting problem for G-shtukas |
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318 | (33) |
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14 Modular forms and Calabi-Yau varieties |
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351 | (22) |
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15 Derivative of symmetric square p-adic L-functions via pull-back formula |
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373 | (28) |
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16 Uniform bounds for rational points on cubic hypersurfaces |
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401 | (21) |
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17 Descent on toric fibrations |
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422 | (14) |
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18 On filtrations of vector bundles over P1z |
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436 | (22) |
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19 On the dihedral Euler characteristics of Selmer groups of Abelian varieties |
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458 | (35) |
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20 CM values of higher Green's functions and regularized Petersson products |
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493 | |
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Luis Dieulefait is Associate Professor in the Department of Algebra and Geometry at the University of Barcelona. D. R. Heath-Brown FRS is Professor of Pure Mathematics at the University of Oxford and has twice been an invited speaker at the International Congress of Mathematicians (ICM). He is one of a growing band of number theorists exploring the interface between analytic number theory and Diophantine geometry, which led him to co-organise the trimester of which this volume is the proceedings. Gerd Faltings is Managing Director of the Max Planck Institute for Mathematics in Bonn. Yuri I. Manin is Professor Emeritus at the Max Planck Institute for Mathematics in Bonn. B. Z. Moroz is Associate Professor at the Max Planck Institute for Mathematics in Bonn. Jean-Pierre Wintenberger is Professor at the Department of Mathematics at the University of Strasbourg.
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